746 CHAPTER 15 KINEMATICS oF RIGID BoDIES: RELATIVE MoTIoN

Example 15.8

An airplane moving at 200 ftlsec is undergoing a roil of 2 rad/min (Fig.15.25). When the plane is horizontal, an antenna is moving out at a speedof 8 ftlsec relative to the plane and is at a position of l0 ft from the cen-terline of the plane. If we assume that the axis of roll corresponds to thecenterline, what is the velocity of the antenna end relative to the groundwhen the plane is horizontal?

A stationary reference XYZ on the ground is shown in the diagram.A moving reference xyz is fixed to the plane with the x axis along the axisof roll and the y axis collinear with the antenna. we announce this for-mally as follows:

Fix ryz to plane.Fix XYZ to ground.

We then proceed in the following manner:

A. Motion of particle (antenna end) relative to xyzto

= 10j ft= 8,/ ftlsec

B. Motion of xyz (moving reference) relative to XyZ (fixed reference)

ft = 200i ftlsec(,) = - $i = _$iradlsec

We now employ Eq. 15.20 to get

,*lfR+toxp8i + Zooi. (-+) x (1oj)

Vxrz = 2Wi + 8j - *ft ft/sec

1oNote that sinca the corresponding axes ofthe references are parallel to each other atthe instant of interest, the unit vectors i,j, and /< apply to either reference at the instant ofinterest. We will arrange.ryz and XyZ this way whenever possible.

Figure 15.25. ryz fixed to plane; XyZfixed to ground.

pVxyz

,r- |

Note from the preceding example that in part A, we are using thedynamics of a particle as presented in Chapters l1-14, while in part B we areimplementing chasles' theorem as presented in this chapter. your authorbased on long experience urges the student to work in this methodical manner.

2radlmin

Example 15.9

SECTION 15.7 RELATIONSHIP BETWEEN VELOCITIES OF A PARTICLE FOR DIFFERENT REFERENCES ,74'7

I an incline with a speed of 10 km/hr in Fig. 15.26. The

L speed a, of 2 rad/sec relative to the tank, apd the gun

red (rotating) at a speed a4of .3 rad/sec relative to the

elocity of point A of the gun barrel relative to the tankround? The gun barrel i$ 3 m in length. We proceed as

.27).

cle relative to xyz

cos 30] + sin 30"k) = 2.60j + 1.50k m

the gun barrel, which has an angular velocity to,

Fix ryz to tufrst.FixXYZtata*,

( do\1ffJ,, = @: X P= (-3i) x (2'60't+ l'5k)

-.780k + .45j m/sec

'elative to XYZ Figure 15.26. Tank with turet and gunbarrel in motion.

R = .65j

rvhich is rotating with angular speed o,

o, XX=2kx65i=@r = 2k rad/sec

te into the basic equation relating V,u.to V*fr.That is,

A tank is moving up an inturret is rotating at a spee(

barrel is being lowered {rr

turret.-What is the relocitand relative to the ground

follows (see Fig. 15.17).

A. Motion of particle

P = 3(cot

Since p is fixed in the

tive to xl'Z, we have

'-.:!

B. Motion of xyz relati

Since R is fixed in tlative to XYZ, we hav

' R=G):

We can now substitu

Vxyz=Vrr+il= (_.180k

+@xp+ .45 j) - r.3i + (2k) x (2.60 j + l.sok)

This result is the desired velocity ofA relative to the tank. Since the tank ts

moving with a speed of (10)(1,000y(3,600) = 2.78 m./sec relative to the

ground, we can say that A has a velocity relative to the ground given as x'Figure 15.27. rrz fixed to turret; XYZfixed to tank.

Yground = V*rr+2'78j

Yg.o*a = -6.5;i+3

. l*'',

65m

-

:

-

::

748 cHAPTER 15 KINEMATICS oF RIcID BoDIF'S: RELATIVE MorIoN

Example 15.10.:X:

i,uA gunboat in heavy'seas is firing its r''rain battery (see Fig' i5 28) The gun

bairel has an anguiar ..,elocity ar, relative to the turret, ivhile the turret has

an angular velocity r0, rel,ative to the ship. If we wish to have the velocity

compirnents of the eirerging shell to be zero in the stationary X and Z

tlirections at a certain specific titne 1. what should a;, and (l). be at this

instant,l At this instant, the ship has a translational velocity given as

V.,i,ip = '02i + 016ft rrVs

Take the inclination of the ban'el to be 0 - '10'. Determine also the veloc-

ity of the gun harrel tiP A.

zZ ,.'3m

-----\:\ A..+ q ::---'--

Figure 15.28. A gunboat in heavy seas firing its main battery

We proceed to solve this problem by the

axes shown on F-ig. 15.28'

following positioning of

Fix x1,z to fulTet.

FixXYZ to the ground (inertial reference).

We can rrr-rri, pt'oCftltl with the detailed anal;'sis of the prolriem'

Y,Y

A, Motion of A relative to x.Yz

p = -(4)(.866U + (4)(.5)k = -3.464i + 2k m

Vo,. = *, x P = al,i x (-3.454i + 2k) = -3'461ark -2a4i rnls

:

ilti.,

,- =.:_

SECTION 15.7 RELATIONSHIP BETWEEN VELOCiTIES OF A PARTICLE FOR DIFFERENT REFERENCES 749

Example 1 5.10 (continued)

B. Motion of xye relative to XyZR = -3jmR = ark x (-3i) + (.02i + .0t6k) - (3a" + .02)i+.ut6/< m/s@ = a2k rad/s

We can now proceed with the calculations.

Vxyz=Y..+R+<oxp

= (*3.464<t& - Za,i)+ (3c0, +.02)j +.016/r + (<rl"f) x (_3.164j + 2k= -3.464orf - 2ari + 3a.i + .O2i + .Ol6k + 3.4&ko^i:. Vxyz = (3rrl, + 3.464a, + .02)i + (-2tr tlj + (_3.464o, + .0161/< n/s

Let(V*rr)* = 0

.'. 6.464a4

Let(.V*rr), = 0

oz = -.003094 radls

or = .004619 rad/s

Finally, we can give V*r, as Yxtz = -.UJ9238j m/s

In some of the homework probrem diagrams, in the remainder of thechapter, a set of axes ,ryz has been shown as a suggestion for use by the stu-dent. This has been done to help clarify the geometry of the diagram. Also. ifthe student chooses to use these axes he/she will be able tr-r **pu.. *n..easily his/her solution with that of the author as presented in the instructor,smanual. However, (and note this carefully) the student must deciire indepen-dently as to how to fix this reference and to state clearly as we have done inthe examples how this reference has beenfixed.

Also, we strongly urge the student to make careful clear diagrams andto follow the orderly progression of steps (A. Motion of particle- etc., etc.followed by B. Motion of rys etc., etc.)

-3.464a, = -.016

-

!

.Jt5.72.-a panicle rotf,tes at a conslant o:q:1n:.,:ottd ot

'urad/sec on a plattortn. rlnrle the plattbrm't1'.:t *it1'1:T::::15,10. A space laboratory, in order to simulate gravitl'' rotates rel-

,:,-.: to inefiial ret'erence XyZ ut ^

rate A)i For occupant A to t'eel

-,,-.,i..n^Ut., what should (0, be ? Cleurly' at the center room B' there

', .l; to zero gravity fo''zt'o-g expentnents .A conveyor along

i: r.f the spokes transports ite;s from the living quarters at the

-":J.;,";h. zero-gravitv laborrtory-at lre cen1e1,

ln paflicular' a

.=i.r. b has a velocity toward B of 5 m/sec relative to the space

.1,,o". Wfru, is its velocity relative to the inertial ret'erence XIZI

Figure P'15'70'

15.i1. Bodies a and & slide away from each other each rvith a

- -,i,sianl velocity of 5 ftlsec along the axis C-C mounted on a plat-

:'n'r. The platlbrm rotates relative to the Sround reference XIZ at

,r. ansular velocity of 10 rad/sec about axis E-E andhas an angular

..;i;;r;l ,aolset' relative to the ground reference XYZ ar

:r-:c trme when the bodies are at a distance r = 3 ft from E-E'

i,.,.*ri* the velocity of particle D relative to the ground reference'

;ffi; ili,,i io,l^or".. aului 1x11e-a, YY-'ft'^,::':::ll'i#;;it* r ",-,l''l'"'"t

the platform " ]l ll',3.llf:#i

:;.TJ;':;:;,;;.;;i'r' r**' an angle or 30' with the )'

axis as shown'l

Z10 rad/sec

i

50 radlsec.+-e-,

\'Pl"tf..,rn.,

Figure P.15.72.

i15.73. A platforn-r A is rotating with constant angular speed {0'

of I rad/sec. A seconcl platftllm B rides on A' contains a rorv of

testtubes,andhasaConstanlangularspeedr0.of.']rad/secrela-tive to the platform A A third piatfor* C is in no u'ay connecteC

*iiit pt",t**. A and B E on plattbrm C is.positioned above A

and B ant.l canies dispensers of chemicals which are electrically

onerated at proper times to dispense drops into the test tubes held

;l';;i,;:;'rr., .in,r'r the angular speed ro' be for platform E

;;;J;;;,to*n ir it is to dispense a drop of chemical having a

)".o ,ung.rri"l velocit.v relative to the test tube below'l

l'igure P'15'73'

15.74, In an amuseinent park ride' the cockpit containing rt'

occupants can rotate at an angular speed t0, relative to the niain tr-

oi.i:n"arm can rotate with angular speed r0' relative to the grou: -

;;; position sirown in the diagram and for 6)' = 2 rad':'

il;; '= j *olr.., find the velocity of point A (coresponc.- 'to the position of the eyes of an occupant) relative to the gror'r:

TE

i

E

Figure P.15.71.

Y-c

-'- '.1-O ttl

-\ 1_)

'.1'.$;76, A tank is moving over rough terrain rvhile llrin_s its rraingun at a fixed target. 'Ihe barrel anci turet oi the gun partly com_pensate for the mcrtion of the tank proper by giving the barrel an

::::,::l-.::r1 o, reirrive to the turiet arrct,-simr:-ltaneously, bygrvlng rne tLtrret xn anurrjrr t,elocitv ro, relative ta the tank propersuch thar anv jnsra,r,i. ,.ru.i,1 ,,r;ir.;;;; ila,rer has ,enrr,elocitl, in the X and Z tlirections rclative to the grounA ref.erence.

What shoulcl rhese an-eular velocities be fbr the fi,ttowing ransla-ti,,tt;ql,','.,,,,,,,1 the trrrk:

VraNr = lOi + .lfr rn/s

15.75. A warer sprinkler has .4 cfs (cubic ftlsec) of warer fedinto the base. The sprinkler turns at the rate f0, of 1 rad/sec. Whatis the speed of the.iet of u,ater relative to the gr.,und at the exits?The outlet area of the nozzle cross section is-.ZS inr. lt'tint; Thevolume of'flow through a cross section is I44. where !.1 is thevelocitl, and A is the area of the cross section. ]

Figure p.15.76.

15.77. We can show that Eq. 15.6 is actually a special case ofUr:.lt*rq For this purpose, consider a rigid bocly moving relariveto XYZ. Choose two points u and l:t ln rne Uo,tyr The body has atranslational velocity con.esponding to the velocity of point a anda rotational velocity co as shown in the diagranr. No* embed areference,r'\.i into the body rvith origin at pciint a. Next. use thisdiagram and consider point D to shor.i that Eq f S.ZO can be refor_mulated ro be identical to Eq. l-5.6.

Figure P.15.77.

Z XYZ stationarv

Figure P.15.74.

Front view

Top view

Figure P.f5.75.

-:

15.78. A simple-impulse type of turbomachine calied a Pelton

water wheel has a singie i"iof *ut"t issuing out of a nozzle and

irnpinging on the system of buckets attached to a wheel' The run-

""i --ni;'t is the assembly of buckets and wheel' has a radius of r

io it "

..,.,,.. of the buckets. The shape of the bucket is also shown

where a horizontal midsection of the bucket has been taken' Note

that the jet is split in two pa(s by the bucket and is rotated relative

,""in.-ir.u.r'in the horizontai plane as rneasured by B' If we

,.*i.., U^rr, and friction' the speed of the waterrelative to the

bucket is unchanged ouring tt.,. action..suppose that 8 liters oi

;;;;;;. second'fln* thro"trgh the nozzle' whose cross-sectional

area at the exit is 2,000 **i tt r = 1 nt' what should tr;' be (in

rpm) for the water on average to have zero^velocity relative to the

grountl in the I direction *i"n it comes ofi the bucket? Take p =

10'. (Why is it desirable to have the exit velocity equal to zero in

the ),direction']) See the hint of Problem 15'75'

-.'\'(0\P'15.8d. A crane moves to the right at a speed of 5 km/hr' The boom

,i;|*nr.n ,rJi - r""r, is being raised at an angular speed r0, reia-

#; ffi;."u "i:.+ ,Ji*.t, *iil" th" cab is rotating at an angular

;;;;I., rad/sec relative to the base' What is the velocitv of pin

B reiative to the $ound at the instant when OB is at an angle of 35'

*i,i, ifr" grornd?-Th" axis of rotation O of the boom is 1 m from the

u*i. of .,ir*.n A-A of the cab, as shown in the diagram'

)il_,,--)rL lrro,"^

AI

<-!} ro,

(,r = .2 rad/sec

ro, = .'l lad/sec

*

$&

qr*

lLl'" r-!ili.

TilL ]I

@v

[-A

Figure P'15'80'

l5.8l.ApowcrsholelnlairlarmACr.otatesrn:ithatrgttlarspee'rrt oi .3 rad/sec relatrve to the cab Arm E'lf rotates at a speed o'

.?.*',;J;. r.rrii'. tt' the main arm AC' The cab rotates abol:

^-t, n-O at a speecl {D, ol '1-5 rad/sec [elative to the tracks whi'"

,,= .orinnru. What is the velocitv cf point I)'.the cer-rtel of ri'

-i-r'.i. ,, th.'inrt"'it ol interest shown in the diagram) All ha'-

length ol 5 rl and 81) has a length of i nt'

-G

I

<,-.]} (D.Figure P'15'78'

15.19. A propeller-driven airplane is moving^ at a specd of 130

km/hr. Also, it is undergoing o y"* rotation of 1/4 rad/sec and is

simultaneously undergoing a-loop rotation of 1/4 rad/sec' lY.11.o:

p"ii.. i, ."",irg at the rate of 100 rprn with a sense in the posltrve

I direction. What is the velocity of the tip of the propeller c rela-

i* ," ,fr" ground at the instant that the plane is horizontal as

shorvn'l The propeller is 3 m in total length and at the instant of

interest the blade is in a vertical position'

l'*A

,<(\/' '///'

z Cb'/orad/sec Ya*

ference rd rad/sec IooP

Figure P.15.79.

Ground reference Iil**$N#*r;k*,,1**'1ffi752

Nozzle

;)&45.)i

V

i ..

))

Figure P.15.81.

15.85. In a merry-go-round, the main platform rotates at thetate at of l0 revolutions per minute. A set of 45" bevel gearscauses B to rotate at an angular speed 0 relative to the platform.The horse is mounted on AB, which slides in a slot at C and ismoved at A by shaft B, as indicated in the diagram, where part ofthe merry-go-round is shown. If AB = 1 ft and, AC = 15 ft,compute the velocity of point C relative to the platform. Then,compute the velocity of point Crelative to the ground. Take g =45" at the instant of interest. What is the angular velocity of thehorse relative ro the platform and relative to the ground at theinstant of interest?

Figure p.15.82.

15.83. A cone is rolling without slipping abour the Z axis suchthat its centerline rotates at the rate 0, of 5 radlsec. Use a mul-tireference approach to determine the total angular velocity of thebody relative to the ground.

m Fieure P.15.83.

tgM.YinO the velocity of gear tooth A relative to rhe groundreference XYZ. Note that cr)l and O, are both relative to theeround. Bevel gearA is free to rotate in the collar at C. Take at, =I radlsec and A)2 = 4 rad/sec.

--TI

200 mmI

+

(b)

Figure P.15.85.

753

@r = 2 rad/sec@z= 4 rad/sec

Figure P.15.84.

15,--.--+

,:.:r....O rruck has a speed Iz ol. l0 n.rilirr anri an acceleration /of 3 mi/hrisec ar tirne r. A cylintler.ol radius .qu"i ,o U it is rollingwithout slipping at rinrc / such thar relative to the truct it has an

..]i.i,:.,:l,rl."U ro, ald aneular accelerarion r,;, oi: ..alr.c anrl jritu/seL--. respectii'el.'r. Determine the l,elocity and acceler.ati.n cfthe center of the cylinder relative to the sr.ound.

Figure p.15.91.

, .tt'

IQ.-YZ:*'A wheel rotatcs with an angular .rpeecl rr;. oi. 5 rad/secrelati,e to a platform. which rotates ",irfr,,p...i r,], ol i0 rad/secrelafive to the ground as shou,n. A collar,rl,r.., J.lrun the spokeof the rvheel, anr.l. when the spt,ke i, rerticl,l ti,. rnttr. t ur .speed of20 filsec. an acceieration of ILr fi/secr ri,rng tt. rpnt".and is positioned I ft fr.om the shaft centerline nf ttre *,t.et. Cu,r_pute the velocity and acceleration of the collar relative to theground at this instant. Fix l.; to platfbrm and use cl,iinclrical coor-dinates.

15.95. In problem 15.j4,fincjthe acceleration ofpointA relativeto the ground.

15.96. In problem 15.79, find the acceleration of the tip of thepropeller relative to the ground reference. Take the yaw rotation tohe zero and rhe loop rotation radius r ro b. ;il';.'

15.97" trn problem 15.g0, find the acceleration ofpoint B relativeIo the ground.

15.98. In problem 15.g0, find the accelerarion ofpoint _B relativeto the ground for the foltowing dara at ttre instlnt of interestshown in the diagram.

@r = .2 rad/secdr =-. 1ra4/sec2@. = .! rad/sectit. = .3 rad/sec2

15.99, In.pr.oblem 15.g2, determine the acceleration of rhe toptip of the gun relative to the grouncl.

15.100. in probleni 15.74. find rhe acceleration ofpoint A rela_tive k) the ground for the configuration shown. fot. r, = 2 rad/secor = 3,rad/sec2,_ rd: = .l .uJl..., ara i. =-2-ratl/sec.. Howmany g's of acceleration is this point subjeci to,?

15.101. In problern 15.g1, find the acceleration of D relative totlre grourrd. lHint; Use two position u..,o., ,o g"i p.]

f5.t02. Find the acceleration of gear tooth A relative to theground in problem 15.g4.

15.103. In problem 15.92, tbe wheel accelerares At the instantunder discussion with 5 rad/sec2 .rl;r. ;; ,h";;;tform, and rhe

lrtto:a*.:"ases irs angular speed at tO ,^J7r#."rative to theground. Find the velocitv and acceleration of ttre coitar relative totire ground,

(t): = -5 radlsec

Figure p.I5.92.

15.93. In Problem 15.71, determine the acceleration ofthe parti-cle at'the instant ofinterest.

,1.r1. In Problem 15.i2, find, the accelerarion of the parricle prelative to the ground reference.

769

z____:==-

with a rate of change of speed of .5 rad/sec2. Find the accelerationvector of the piston head relative to the ground.

-rI

I

.6 rr.r

I

Figure p.15.111.

15.112. A conrmunications satellite has the Ibllorvins morionrelative to an inertial ref'erenc&XyZ.

or = 3i + 4j + 10/i rad/s

Yr=ao=0

15,113, A particle moves in a slot of a gear with speed V = 2m/s and a rate of change of speed i7 = 1.2 rn/s2 both r.elative tothe _qear. Find the acceleration vector for the particle at the config-uration shown relative to the ground reference XyZ.

V = 3 m/secV = .4 m/sec2

a)r = .34 rad/sec(O, = .$ 1u67.""bz= .5 rad/sec2

REL TOTUBE

REL TO AREL TO

GND.

r =.3 rnI =.7nt

A wheel atA is rotating relative to the satellite at a constant speed ,r.= 5 rad/s. What is the acceleration of point D on the wheel ielativeto XfZ at the instant shown? The followine additional clata appl1,:

0A = lrn AD = .2m

Y = 2 m/s relatil'e to gear\/ = 1.2 m/sl relati,,'e rtl searo), - .(18 r'adisrr;, = .()2 rlrd/sre =.15 nr

Figure p.15.113.

15.114. A submarine is undergoin-t an evasive maneuver. At theinstant of interest, it has a speed ll = t0 rn/s and an acceleration(?! = l5 m/sl iil its cellter 01-mass. It also has an angulnr velocitvabout its center oi rnass Cl ol 1,;, = .-5 ratl/s .n,1 ,n .=,igula, *cc.i-eratiort ri, = .02 rad/s:. insitle is a oart of arr inertiai guiclancesysten) thrit consists o1' a wheel spinnine *,ith speed o., = 20rad/s aboul a vertical axis ofrhe ship. A!ong a spoke sho*,n at theinstant of interest. a particle is rntlvin_u tou,ard thr: ccnter *,ith r, r:and i:as siven in the diagr-am. lVhar is the acceieration of the par_ticle relative to inertial rcfcrence .{)'z'l Just rvriie.ut rhe rbrmula-tions tbr a.rrrblfi do not carry out the cross prrrrJucts.

dt = 2i + 3t radlsl

771

Wheel is in theXIl plane at time ,

Figure P.15.112. Figure P.15.l14.

15.115. Find Y and a ofB relative to XyZ. The single blade por_tion AB is rotating as shown relative to the helicopter with speed a;,about an axis parallel to the X axis at the instant of interest olhite thientire blade system is rotating about the vertical axis with speed a-l,.

15.116. An F-16 fighter plane is moving at a consranr speed of800 km/trr while undergoing a loop as shown in rhe diagram. Atthe instant of interest, it has an angular roll velocity r,.r, of 5rad/min relative to the ground. A solenoid is activated it thisinstant and moves the moving portion of a gate valve downwartl ata speed of 3 m/s with an acceleration of 5 m/s2 all retative to theplane. What is the total external force on the moving portion ofthe,gate valve if it is 1 m above the axis of roll at the instant ofinterest? The moving portion of the valve has a weight of j 0 N.

i0km i

"t.. \r''\'- ,'-' z\-,N\ ' I-:G-

% -i-,x'

: Figure P.15.116.

15.1]7. A robot moves a body held by its ,Jaws" G as shown inthe diagram. What is the velocity and acceleration of point A atthe instant shown relative to the ground? Arm EH is welded to thevertical shaft MN. Arm HKG is one rigid member which rotatesabout EII. How do you want to fix ryz?

772

a- co^

I

o -l'-

fl,,=?::fl:::, ] Rerutiu" to sround

?,:= i;fl:::, ] n"tuti'" to arm EH

Figure P.15.117.

15.118. The tunet of the main gun of a destroyer has at time t anangular velocity at = 2 radls and a rate of change of angularvelocity 6r = 3 rad/s2 both relative to the ship. The gun barrelhas co, = .5 rad/s and ti, = .3 radls2 relative to the turret.

(a) Find the acceleration of the tip A of the gun at time ,relative to the destroyer.

(b) If the destroyer has a translational acceleration rela-tive to land equal to

. a- = .0!i + .26j - 2.2k m/s2I Deslroler '-a' ''"J

what is the acceleration of A relative to the land at time r?

*

n+Z

I

I

r-)-'Ground reference

d =l0mY = l30km/hr''

, V = lOkm/hr/sec@r = 100 rpm(0' = 3 radls

Figure P.15.115.

ff

.p

I

r:a'- :'. y,.,.r.rt,r..rtr,r,rr-,::,,i,r,:11:

@r=2radlsecdr=3rad/sec2

@: = '5 rad/sec

dz = .3 rad/sec2

z

.]r9

Figure P.15.118.

a'-.#

*